First of all you put "hw" in the title, so why didn't you tag it with "homework"? And second of all it's very impolite to post in the imperative - you're not writing a textbook nor are you a teacher assigning homework
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kahenFeb 27 '11 at 5:25

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What does it mean for $A$ to be bounded? I.e., how would you write that out explicitly?
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Jonas MeyerFeb 27 '11 at 5:26

Definition:A metric space $M$ is called bounded if there exists some number $r>0$, such that $d(x,y) \leq r$ for all $x, y \in M$.

Since you are assuming $A$ to be bounded then there is some number $k$ such that $d(x,a) \leq k$ for all $x,a\in A$. Since $B \subset A$ and this condition happens for all elements in $A$, therefore it should happen for elements in $B$ as well.